Monday, May 1, 2023

Does my existence by itself confirm a multiverse?

Suppose I am considering two hypotheses, H1 and H2, and according to H2 there are more people. Does the fact that I exist give me reason to prefer H2, all other things being equal? If so, then my existence is apt to confirm the existence of a multiverse over a single universe.

Here is one reason to think this works. The probability that I exist in a given world, all other things being equal, seems proportional to the number of people in that world. Each person in that world corresponds to another opportunity for me to exist.

While this is tempting, here is a toy model that should give us pause. Suppose that I am defined by a real number parameter between 0 (inclusive) and 1 (not inclusive). According to hypothesis H1, a single real number is picked uniformly at random in the range, and the person with that parameter is created. According to hypothesis H2, two real numbers are picked uniformly and independently in the range, and persons corresponding to these are created. Learning that a person with my parameter is created seems to provide me with evidence for H2, since it’s twice as likely on H2 as on H1.

But this is tricky. In classical probability theory, it is correct to say that my parameter is twice as likely to be generated on H2 as on H1, but that’s only because both probabilities are zero, and zero is twice zero, so while H2 is twice as likely as H1, it is also true that H1 is twice as likely as H2!

Perhaps, though, we want to depart from classical probability theory in some way, say by allowing non-zero infinitesimal probabilities or by an intuitive handwavy “this is twice as likely as that”. However, it is then no longer clear that on H2 there is twice as big a chance of hitting my parameter. For there are (infinitely) many ways of picking a number between 0 and 1 uniformly randomly.

Here’s one way:

1. You write down “0.”, then roll a fair ten-sided die infinitely many times, writing down the results as the digits after the decimal point, thereby generating a decimal representation of a number. If the number ends with infinitely many nines, try again.

(The final proviso is to ensure that intuitively each number is equally likely. Without that proviso, 1/10 would be more likely than 1/3, as there would be two ways of getting 1/10, namely 0.1000... and 0.0999..., but only one way to get 1/3, namely 0.3333.....)

Here is another way:

1. You write down “0.”, then roll a fair ten-sided die infinitely many times, omitting the results of the first die throw, but writing down the results as the digits after the decimal point, thereby generating a decimal representation of a number. If the number ends with infinitely many nines, try again.

Intuitively, method B has ten times the probability of generating any given number than method A has, as long as literally the numerically same die throws occur in the two cases. For consider the number 1/3 = 0.3333.... By method A to generate it you need every die to show a three. By method B, to generate 1/3, all you need is for all the die throws other than the first one to be threes, and so there are ten times as many ways to generate the number.

Now, if the single selection of a parameter on H1 uses method B while the double selection of a parameter on H2 uses method A, then intuitively we are five times as likely to generate my parameter on H1 than on H2. Thus merely saying that on both hypotheses the parameters are generated uniformly is insufficient to determine how the comparison between the probabilities of generating my parameter goes.

We might insist that in both hypotheses the same method for generating parameters is used. But notice that in cosmological applications, this is implausible. If H2 is some multiverse hypothesis and H1 is a single universe hypothesis, we are unlikely to be able to count on the two hypotheses involving even the same laws of nature, much less the same selection process for the parameters of the persons. (Besides all this, it is really unclear what it even counts to say that there are two different runs of method A.)

So, here’s what I am thinking. On classical probability theory, there is no difference in the probability of my parameter getting generated on H2 than on H1, because both probabilities are zero. On non-classical probability theory, we can perhaps make sense of a difference between the probabilities, but cannot count on the hypothesis with more people being more likely to generate my parameter.

Given all this, there does not seem to be a way of making sense of comparing the evidential impact of my existence on the two hypotheses using probabilistic methods. Maybe all we have is intuition.